Symbolic Closed-Form Fibonacci

Let $V := \{(a_j)_{j\in\mathbb{N}}\subset\mathbb{C}|a_n=a_{n-1}+a_{n-2}\forall n>1\}$ be the two-dimensional complex vector space of sequences adhering to the Fibonacci recurrence relation with basis $B := ((0,1,\dots), (1,0,\dots))$.
Let furthermore $f: V\to V, (a_j)_{j\in\mathbb{N}}\mapsto(a_{j+1})_{j\in\mathbb{N}}$ be the sequence shift endomorphism represented by the transformation matrix

$A := M^B_B(f) = \begin{pmatrix}1&1\\1&0\end{pmatrix}$.

By iteratively applying the sequence shift a closed-form solution for the standard Fibonacci sequence follows.